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For fixed $n$ and $k \le n$ the distance $|X_k-Y_k|$ is uniformly distributed in $[0,1/(2n)]$. Thus if $\alpha_n<1/(2n)$, then $$P\left[\max_{1 \le k \le n}|X_k-Y_k| \le \alpha_n \right]=(1-2n\alpha …

There is a symmetry argument based on uniform spanning trees. If you ask for the effective resistance across an edge $e$ in the discrete torus torus $G_n=[-n,n]^2$ with periodic boundary conditions, i …

The requested estimate follows from a theorem of Kesten [2] about concentration functions. See in particular the inequality (1.6) on page 135 of [2]. Taking $L=\lambda<1$ in this inequality gives the …

This question would be more suitable for MSE. Indeed the expectations tend to infinity. This follows from Fatou’s lemma. All you need are the first and fourth assumptions. The second and third assumpt …

Alas, no such inequality can hold. Suppose that the symmetric $X_i$ take values $\pm 1$ and $t=n-1$. Then $$ \Pr[\max_{k \in [n]} |\sum_{i \in [n] \setminus \{k\}} X_i| \ge t] =(n+1)2^{1-n} $$ but $ …

This question has been studied extensively in the computer science literature under the name "balls in bins"; see [1] which gives quite tight bounds in Theorem 1, page 161 and also describes prior wo …

In the general setting you propose, the answer is negative. Suppose that $X_i$ take the values $\pm 1$ with equal probability. Let $h: {\mathbb R} \to {\mathbb R}$ be a Lipschitz function with consta …

The expected length needed to see all patterns of length $k$ in an alphabet of size $a$ is $k \ln(a) a^k (1+o(1))$. See Propositions 11.9 and 11.10 in [1] for the case $a=2$. [1] Levin, David A., and …

It is amusing that a unit flow of finite energy in the $d$-dimensional lattice for $d \ge 3$ can be easily constructed from Polya's urn; see ``Polya's theorem on random walks via Polya's urn'', Amer …

The canonical argument proving the upper bound is in the comment by Mathworker21, which has now been posted as an answer. I just want to add that the constant $e$ obtained there is sharp. Indeed, if …

Alas, the general bound you are hoping for cannot hold. Take $X_0:=1$ and continue the sequence as a simple random walk with probability $1/k$ and as the all ones sequence with probability $1-1/k$. T …

First suppose that hypotheses (a) and (b) hold. I will write $w_{i,n}$ instead of $w_i^n$ for clarity. Given $\epsilon>0$, the Monotone convergence theorem implies that there exists $M$ such that $ …

Shishir, this is quite elementary: write the probability $P_2(\omega)$ of any individual bit sequence $\omega$ as $P_1(\omega) f(\omega)$ where by definition, $f(\omega)=\exp(-\hat{kl}_n)$. Finally, s …

Expanding the comment above to a complete proof of Noam Elkies' formula: Let $\tau$ be the first meeting time for independent simple random walks $X_t, Y_t$ and $Z_t$ started at the even points $x<y …

A sufficient condition to have $Y⊥D|Z$ is that $f$ is injective. The sharp condition (if $Y$ and $D$ are not specified) is $(*)$ $\sigma(X)$ should be contained in the completion of $\sigma(Z) …